Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C2×C3 | (C3⋊D3)⋊A4 |
| commutator, Fitting, cosocle, hypocenter | Q8×H(3,𝔽3) | C2×C3 |
| Frattini | C2×C32 | D3×A4 |
| prime radical | C2×#272 | A4 |
| LCS | G ▷ Q8×H(3,𝔽3) | |
| UCS | C1 ◁ C2×C3 | |
120 conjugacy classes of subgroups.
| transitive degree | μt(G) = 216 |
Polycyclic presentation:
2,3,2^2,3^2,2,3|b^3=h^2,c^2=g,d^2=g,e^a=e^2,f^a=f^2h,c^b=cd,d^b=c,e^b=ef,d^c=dg,f^e=fh54 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 765);